Distributions: normal and skewed distributions; characteristics of normal…

The idea

A normal distribution is the symmetrical bell curve occurring when most people score near the middle and fewer score towards either extreme; height, shoe size and IQ approximate it. Its characteristics are: symmetrical about the midpoint, the mean, median and mode all falling at the same central point, and tails that stretch towards the horizontal axis without ever touching it, as extreme scores remain possible. Roughly 68% of scores sit within one standard deviation of the mean. A skewed distribution is asymmetrical and named after its tail. Positive skew has a long tail to the right, scores bunched at the low end and the mean pulled above the median and mode, as seen in a very difficult test; negative skew is its mirror image.

What earns the marks

• Normal distribution described as symmetrical and bell-shaped, with most scores concentrated around the mean.
• In a normal distribution the mean, median and mode all fall at the same central point.
• The tails approach but never meet the horizontal axis, so extreme scores are rare rather than impossible.
• Skewed distribution defined as asymmetrical, with the name taken from the direction of the long tail.
• Positive skew: long tail to the right, scores clustered at the lower end, mode lowest and mean highest.
• Negative skew: long tail to the left, scores clustered at the upper end, mean lowest and mode highest.
• An example that explains the shape, such as a test that was too hard for the sample producing positive skew.

Common mistakes and how to put them right

• Naming the skew after where the bulk of scores sit rather than after the tail, which reverses positive and negative every single time.

• Drawing a skewed curve but leaving the three averages together in the middle, when the whole point of the diagram is that they separate.

• Writing that the tails touch the horizontal axis, which contradicts the idea that extreme scores remain possible.

• Incorrectly citing reaction times as normally distributed, when they are typically positively skewed.

Exam tips

• Sketch it. A labelled curve with the mean, median and mode marked in the right order earns marks that prose struggles to.
• If you cannot recall which skew is which, draw the tail first and read the name off the side it points towards.
• Where a stem hands you a mean well above the median, that is the examiner signalling a positively skewed distribution.